Binary, IPv4, and Subnets

The IPv4 protocol which we broadly (but not totally) use today rests on an addressing system that was designed in the 1970s and formally published in 1980. It uses quad dotted 32 bit addresses, which can be written in various notations, and a straightforward subnetting scheme.

Binary

For example, the address 1.0.0.1, which belongs to CloudFlare, is constituted by four separate octets, 1, 0, 0, and 1. We can convert these numbers into alternative representations like binary, which is a number N base-2, N₂. And octets here are simply integers ranging from 0 to 255. In binary, we represent numbers like this by powers of 2. If this doesn't make sense, I'll explain below:

0 = 0 = (2^0 * 0)
1 = 1 = (2^0 * 1)
2 = 10 = (2^1 * 1) + (2^0 * 0)
3 = 11 = (2^1 * 1) + (2^0 * 1)
4 = 100 = (2^2 * 1) + (2^1 * 0) + (2^0 * 0)
5 = 101 = (2^2 * 1) + (2^1 * 0) + (2^0 * 1)
6 = 110 = (2^2 * 1) + (2^1 * 1) + (2^0 * 0)
7 = 111 = (2^2 * 1) + (2^1 * 1) + (2^0 * 1)
8 = 1000 = (2^3 * 1) + (2^2 * 0) + (2^1 * 0) + (2^0 * 0)

Decimal to Binary

To find the binary of a number like 3, we divide by 2 and note the remainder each time. First, 3 ÷ 2 gives a remainder of 1, then 1 ÷ 2 gives another remainder of 1. Reading the remainders from bottom to top, we get "11" as the binary representation of 3.

Binary to Decimal

But how do we convert a binary representation of three back to decimal form? The number three is "11" in binary representation.

So we calculate by powers of 2 from our left most significant bit N, to the right least significant bit, which is 0, e.g. from 2^N to 2^0. And each time, we multiply this value by the binary value, 0 or 1, of each individual digit, like this. Then we take the sum of this sequence. For example, for the binary representation of three, which is "11":

(2^1 * 1) + (2^0 * 1) = 2 + 1 = 3

And for the binary representation of two, which is "10", we do this:

(2^1 * 1) + (2^0 * 0) = 2 + 0 = 2

And in software, we expand this notion and use the 8 bit standard to construct a byte representation our possible octet values from 0 to 255. And we write them like this:

0 = 00000000 
1 = 00000001
2 = 00000010
..
254 = 11111110
255 = 11111111

32-bit addressing

Our first digit in CloudFlare's DNS IPv4 address 1.0.0.1 is represented by 1₂. And 0, by 0₂. This format follows for any other numbers in an IP address. And altogether, we can rewrite the address 1.0.0.1 in binary like so:

00000001.00000000.00000000.00000001

We say this binary represents an address within the 32-bit IPv4 addressing system. That is, each IP address is composed of four 8-bit octets. And each bit can be 1 or 0. Therefore, we say there are 2³² possible IPv4 addresses, or approximately 4.3 billion possible unique addresses.

IPv4 Subnetting

Subnets are a strategy for dividing a particular domain into subnetwork blocks. This involves what we call a "subnet mask."

But to get a better understanding and visualize it, each IP range, e.g. /30, etc - delineates a range of addresses for a particular routing table. And each imply a subnet mask, like this:

/0 - 0.0.0.0 
/1 - 128.0.0.0
/2 - 192.0.0.0
/3 - 224.0.0.0
/4 - 240.0.0.0
/5 - 248.0.0.0
/6 - 252.0.0.0
/7 - 254.0.0.0
/8 - 255.0.0.0 (Class A)
/9 - 255.128.0.0
/10 - 255.192.0.0
/11 - 255.224.0.0
/12 - 255.240.0.0
/13 - 255.248.0.0
/14 - 255.252.0.0
/15 - 255.254.0.0
/16 - 255.255.0.0 (Class B)
/17 - 255.255.128.0
/18 - 255.255.192.0
/19 - 255.255.224.0
/20 - 255.255.240.0
/21 - 255.255.248.0
/22 - 255.255.252.0
/23 - 255.255.254.0
/24 - 255.255.255.0 (Class C)
/25 - 255.255.255.128
/26 - 255.255.255.192
/27 - 255.255.255.224
/28 - 255.255.255.240
/29 - 255.255.255.248
/30 - 255.255.255.252
/31 - 255.255.255.254 (Used for point-to-point links)
/32 - 255.255.255.255 (Single IP address)

A subnet mask is a bitmask that denotes how many bits are used for networking and which are for hosting.

For example, the subnet mask for the /30 block is 255.255.255.252. In binary, this address converts to the values:

11111111.11111111.11111111.11111100

This particular subnet mask assigns 30 bits to network addressing, leaving us only 2 bits for our own possible host addressing. Because IPv4 is a 32 bit system.

And we're using base-2, so we take the 2 leftover bits for hosting and raise it to the number of leftover bits we have, which in this case is 2.

So we say 2² leaves us with a resulting 4 possible host addresses.

For example, we can take our IP address 1.0.0.1, and subnet mask 255.255.255.252, and perform a bitwise AND operation:

00000001.00000000.00000000.00000001

11111111.11111111.11111111.11111100

-------

00000001.00000000.00000000.00000000

We have taken the IP address and subnet mask, and compared the binary of each corresponding bit. In an AND operation, if we have 0 and 0, the result is 0. If we have 1 and 0, the result is 0. But if we have 1 and 1, the result is 1. Here, this gives us the network address of the range. It produces the binary output equivalent to 1.0.0.0, which is our base network address.

And following our rule for calculating the host addresses using the leftover bits we have, we say we have 2², or 4 IPv4 addresses, including our network address: 1.0.0.0, 1.0.0.1, 1.0.0.2, and 1.0.0.3.

The lower address, 1.0.0.0, is the network address, while the upper address, 1.0.0.3, denotes the broadcast address. The two addresses between, 1.0.0.1 and 1.0.0.2, denote the actual usable host range.

If we were calculating 1.0.0.1 with the /29 block, subnet 255.255.255.248, we would have 29 bits set to network addressing and 3 bits for our own host addressing. So we would again take 2 and raise it to the number of free bits we have: 2³, which gives us 8. Hence, the 8 addresses, with our actual usable range in between the network and broadcast addresses:

1.0.0.0 (Network address)
1.0.0.1
1.0.0.2
1.0.0.3
1.0.0.4
1.0.0.5
1.0.0.6
1.0.0.7 (Broadcast address)